The definition of the return time (p. 117) and the beginning of the proof of Proposition 8.3 of our paper in Vol. 35 of Ann. Scient. Ec. Norm. Sup. (2002) are not correct. We give an amended version which shows that none of the statements are affected. We take this opportunity to correct some other mistakes (without consequences), e.g. in Sublemma 7.2(3) and Lemma 7.10. Published by Editions scientifiques et medicales Elsevier SAS

It has been known since the pioneering work of Jakobson and subsequent work by Benedicks and Carleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure. Young and Keller-Nowicki proved exponential decay of its correlation functions. Benedicks and Young [8], and Baladi and Viana [4] studied stability of the density and exponential rate of decay of the Markov chain associated to i.i.d. small perturbations. The almost sure statistical proper-ties of the sample stationary measures of i.i.d. itineraries are more difficult to estimate than the averaged statistics. Adapting to random systems, on the one hand partitions associated to hyperbolic times due to Alves [I], and on the other a probabilistic coupling method introduced by Young [26] to study rates of mixing, we prove stretched exponential upper bounds for the almost sure rates of mixing.

Given a number field K=Q and a polynomial P ∈ Q[t], all of whose roots are in Q, let X be the variety defined by the equation NK(x) D P(t). Combining additive combinatorics with descent we show that the Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation on any smooth and projective model of X.

In [11], A Givental introduced a group action on the space of Gromov-Witten potentials and proved its transitivity on the semi-simple potentials In [24, 25], Y-P Lee showed, modulo certain results announced by C Teleman, that this action respects the tautological relations in the cohomology ring of the moduli space (M) over bar (g, n) of stable pointed curves Here we give a simpler proof of this result In particular. it implies that in any semi-simple Gromov-Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the geometric Gromov-Witten potential coincides with the potential constructed via Givental's group action As the most important application we show that our results suffice to deduce the statement of a 1991 Witten conjecture relating the r-KdV hierarchy to the midsection theory on the space of r-spin structures on stable curves We use the fact that Givental's construction is, in this case, compatible with Witten's conjecture, as Givental himself showed in [10]

5. Favre, Charles

et al.

Jonsson, Mattias

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). University of Michigan, United States .

We study the dynamics in C-2 of superattracting fixed point germs and of polynomial maps near infinity. In both cases we show that the asymptotic attraction rate is a quadratic integer, and construct a plurisubharmonic function with the adequate invariance property. This is done by finding an infinitely near point at which the map becomes rigid: the critical set is contained in a totally invariant set with normal crossings. We locate this infinitely near point through the induced action of the dynamics on a space of valuations. This space carries an R-tree structure and conveniently encodes local data: an infinitely near point corresponds to an open subset of the tree. The action respects the tree structure and admits a fixed point-or eigenvaluation-which is attracting in a certain sense. A suitable basin of attraction corresponds to the desired infinitely near point.

Let M be an m-dimensional differentiable manifold with a nontrivial circle action S = {S-t}(t is an element of R), St+1 = S-t, preserving a smooth volume mu. For any Liouville number alpha we construct a sequence of area-preserving diffeomorphisms H-n such that the sequence H-n circle S-alpha circle H-n(-1) converges to a smooth weak mixing diffeomorphism of M. The method is a quantitative version of the approximation by conjugations construction introduced in [Trans. Moscow Math. Soc. 23 (1970) 1]. For m = 2 and M equal to the unit disc D-2 = {x(2) + y(2) <= 1} or the closed annulus A = T x [0, 1] this result proves the following dichotomy: alpha is an element of R \ Q is Diophantine if and only if there is no ergodic diffeomorphism of M whose rotation number on the boundary equals alpha (on at least one of the boundaries in the case of A). One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if alpha is Diophantine, then any area preserving diffeomorphism with rotation number alpha on the boundary (on at least one of the boundaries in the case of A) displays smooth invariant curves arbitrarily close to the boundary which clearly precludes ergodicity or even topological transitivity.